Lesson

L E S S O N 1.8

When curiosity turns to serious matters, it’s called research. MARIE VON EBNER ESCHENBACH

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Space Geometry

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L esson 1.1 introduced you to point, line, and plane. Throughout this chapter you have used these terms to define a wide range ofother geometric figures, from rays to polygons. You did most ofyour work on a single flat surface, a single plane. Some problems, however, required you to step out ofa single plane to visualize geometry in space. In this lesson you will learn more about space geometry, o solid geometry.

S aceall points. Unlike onedimensional lines and twodimensionalis the set of lanes, space cannot be contained in a flat surface. Space is threedimensional, or “3D.”

Let’s practice the visual thinking skill of presentingthreedimensional (3D) objects in twodimensional (2D) drawings.

The geometric solid you are probably most familiar with is a box, o rectangular prism. Below are steps fo making a twodimensional drawing of arectangular prism. This type o drawing is called anisometric drawin .It shows three sides ofan object in one view (an edge view). This method works best with isometric dot paper. After practicing, you will be able to draw the box without the aid of the dot grid.

Use dashed lines for edges that you couldn’t see ifthe object were solid.

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C linde

Cone

Prism

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The threedimensional objects you will study include the six types ofgeometric solids shown below.

The shapes ofthese solids are probably already familiar to you even ifyou are not familiar with their proper names. The ability to draw these geometric solids is an important visual thinking skill. Here are some drawing tips. Remember to use dashes for the hidden lines.

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P ramid

S here

Hemis here

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Solid geometry also involves visualizing points and lines in space. In the following investigation, you will have to visualize relationships between geometric figures in a lane and in space.

S aceGeometr Step 1 Make a sketch or use physical objects to demonstrate each statement in the list below. Step 2 Work with your group to determine whether each statement is true or false. If thestatement is false, draw a picture and explain why it is false. 1. For any two points, there is exactly one line that can be drawn through them. 2. For any line and a point not on the line, there is exactly one plane that can contain them. 3. For any two lines, there is exactly one plane that contains them. 4.two coplanar lines are both perpendicular to a third line in the same If lane, then the two lines are parallel. 5. Iftwo planes do not intersect, then they are parallel. 6.two lines do not intersect, then they are parallel. If 7. Ifa line is perpendicular to two lines in a plane, and the line is not contained in the plane, then the line is perpendicular to the plane.

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EXERCISES For Exercises 16, draw each figure. Study the drawing tips provided on the previous age before you start. 1.Cylinde 2.Cone 3.Prism with a hexagonal base 4.Sphere 5.Pyramid with a heptagonal base 6.Hemisphere 7.The photo at right shows a prismshaped building with a pyramid roofand a cylindrical porch. Draw a cylindrical building with a cone roofand a prism shaped porch. For Exercises 8 and 9, make a drawing to scale ofeach figure. Use isometric dot paper. Label each figure. (For example, in Exercise 8, draw the solid so that the dimensions measure 2 units by 3 units by 4 units, then label the figure with meters.)

8.A rectangular solid 2 m by 3 m by 4 m, sitting on its iggest face. 9.A rectangular solid 3 inches by 4 inches by 5 inches, resting on its smallest face. Draw lines on the three visible surfaces showing how you can divide the solid into cubicinch boxes. How many such boxes will fit in the solid? For Exercises 1012, use isometric dot paper to draw the figure shown. 10. 11.12.

Anet is a twodimensional pattern that you can cut and fold to form a threedimensional figure. Another visual thinking skill you will need is the ability to visualize nets being folded into solid objects and geometric solids being unfolded into nets. The net below left can be folded into a cube and the net below right can be folded into a pyramid.

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13.Which net(s) will fold to make a cube?

For Exercises 1417, match the net with its geometric solid. 14. 15.16.

17.

When a solid is cut by a plane, the resulting twodimensional figure is called asection. For Exercises 18 and 19, sketch the section formed when each solid is sliced by the plane, as shown. 18. 19.

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All of the statements in Exercises 2027 are true except for two. Make a sketch to demonstrate each true statement. For each false statement, draw a sketch and explain why it is false.

20.Only one plane can pass through three noncollinear points.

21.If aline intersects a plane that does not contain the line, then the intersection is exactly one point.

22.If twolines are perpendicular to the same line, then they are parallel.

23.different planes intersect, then theiIf two intersection is a line.

24.line and a plane have no points in common, then they are parallel.If a

25.plane intersects two parallel planes, then the lines of intersection are parallel.If a

26.If threeplanes intersect, then they divide space into six parts.

27.If twolines are perpendicular to the same plane, then they are parallel to each other.

Review 28.If thekiteDIANwere rotated 90° clockwise about the origin, to what location would pointe relocated?

29.Use your ruler to measure the perimete ofWIM(in centimeters) and you rotractor to measure the largest angle.

30.Use your geometry tools to draw a triangle with two sides of length 8 cm and length 13 cm and the angle between them measuring 120°.

ual Distances Here’s a real challenge: Show four points, B, C,andDso thatB=C=C=D=D=CD.

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